# Full text of "A Scale for Measuring Pupils' Ability to Demonstrate Geometrical Theorems"

## See other formats

```STOP

Early Journal Content on JSTOR, Free to Anyone in the World

the world by JSTOR.

Known as the Early Journal Content, this set of works include research articles, news, letters, and other
writings published in more than 200 of the oldest leading academic journals. The works date from the
mid-seventeenth to the early twentieth centuries.

We encourage people to read and share the Early Journal Content openly and to tell others that this
resource exists. People may post this content online or redistribute in any way for non-commercial
purposes.

journal-content .

JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people
discover, use, and build upon a wide range of content through a powerful research and teaching
platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit
contact support@jstor.org.

A SCALE FOR MEASURING PUPILS' ABILITY TO
DEMONSTRATE GEOMETRICAL THEOREMS

J. H. MINNICK

University of Pennsylvania

It is desirable that a scale shall be developed for the
measurement of each of those abilities upon which the mastery
of geometry is dependent. Among these is the ability to
demonstrate a theorem when the figure is drawn, and the
hypothesis and conclusion are given. The chief difficulty in
developing a scale for measuring this ability is the time element.
The writing of a complete demonstration requires too much
of the pupil's time, and the scoring of such a scale is time
consuming. On a former occasion 1 , the writer used a geometry
test in which the drawing for a theorem was given and the
hypothesis and conclusion were stated. The pupil was then
the theorem. Although the exercises of this test were poorly
selected, the results were so favorable that it seemed possible*
by the use of this method, to construct a satisfactory scale.
This scale would have the advantages of requiring no writing
on the part of the pupil and of being easily scored. The
following is a report of an attempt to construct such a scale.

From various sources eleven exercises, apparently suited
to the purpose of the scale, were selected. The time available
for giving the test made it impossible for each pupil to solve
all the exercises. In order to get a random selection of pupils
solving each exercise, the following plan was used: Each
exercise was printed on a separate sheet of paper in the manner

1 J. H. Minnick. An Investigation of Certain Abilities Fundamental to the Study of
Geometry.

102 THE SCHOOL REVIEW

shown on pages 107-109. The exercises were then assembled in
groups of three each in such a way that every possible com-
bination of the exercises was taken, any one combination
occurring as often as any other.

The following directions were sent to each teacher who
gave the tests:

1. See that each pupil is supplied with a pencil and ruler.

1. Read to the pupils: I am going to give you some geometry exercises.
In order that you may all have the same chance, I want you to start at the
same time. Do not open the set of questions which you are about to receive
until I give the signal to begin work by tapping on the desk.

3. Distribute the questions.

4. Have pupils fill out blanks on the cover sheets of their questions.

5. Read to the pupils: At the top of each sheet which you have received
there is a geometrical figure. Below this figure there is a statement of what
is given and what is to be done. When the signal to begin work is given,
fold back the cover sheet, read carefully what is given and what is to be
proved in one of the exercises, and then make any additional drawings
that are necessary to prove the exercise. Thus, if, in the triangle ABC
AC = BC [place drawing on the board] and if we are to prove that Z.A =
LB, we may draw CD to the mid-point of AB. When you have completed
this exercise proceed in a similar way to do the other exercises. Do not
write any explanation on your papers and do not prove the exercises. All
I want to know is whether you can make the correct drawings. You may
do the exercises in any order you prefer. When you have completed the

6. Give the pupils a chance to ask questions concerning the instructions,
but do not reveal the contents of the questions by your answers.

7. Note the time and then give the signal, thus: "Ready," and tap
on the desk with your pencil.

8. Do not permit talking in the class until all papers have been returned
to you.

9. In the case of any irregularity on the part of any pupil during the
test, make a note on the cover sheet of his questions indicating the exact
nature of the irregularity.

10. As the pupils hand you their papers, note the amount of time
required by each, and record it on the upper right-hand corner of the cover
sheet of his or her paper.

PUPILS' ABILITY TO DEMONSTRATE THEOREMS

103

The tests were given in thirty high schools distributed
throughout the country and ranging in size from a few hundred
pupils to several thousand. Each of the eleven exercises was
solved by at least seven hundred pupils. These pupils had
completed either the first two books of plane geometry or all
of plane geometry.

TABLE I

Percentage of Pupils Solving Each Exercise Correctly

Number of
Pupils

Exercises

50
ICO

150
200
250
300

35°
400
450
500

55°
600
650

700

42
37
34
34
33
35
38
38
41
41
41
42
42
42

76

70

7i

72

75
76
77
77
77
79
81
81
82
82

20
28
33
37
42
45
49
46
46
48
49
49
49
49

42
42
38
38
37
37
37
38
36
35
3°
29
28
28

68
62

63
64
64
66

67
68
66
66
66
66

67
68

28
20
22
26
28

30
32
33
35
34
32
32
3i
3i

68
60
60
64
60
61
64
64
62
60
59
58
58
57

14
12

l 3

'5

12

J 5

14
H
14
14
13
13

40
41
41
44
45
45
45
43
44
42
42

41
40

39

16

24
40
46

43
41
36

32
28

25
23
22

19

20

46

47
36
36
36
37
38
36
37
36
36
35
34
34

In order to secure uniformity of marking, all papers were
marked by the author. Any drawing which made a proof
possible was usually counted as a correct solution. If, how-
ever, in addition to the correct lines, unnecessary lines were
drawn, the solution was marked as incorrect. Also, if the
proof resulting from a drawing seemed to be too difficult for
high-school pupils, those pupils making the drawing were asked
to complete the proof. If the majority could not do so, the
drawing was not accepted as a correct solution. A careful

io 4 THE SCHOOL REVIEW

record of all acceptable drawings has been kept and will be
included in the "Directions for Scoring Papers 1 ".

Table I gives the percentage of pupils solving each exercise
correctly. The table is arranged in a cumulative way, each
new line including the data for fifty additional pupils. Thus
of the first fifty pupils solving Exercise i, 42 per cent got it
correct; of the first hundred solving Exercise i, 37 per cent
got it correct, etc. The table shows that the number of pupils
tested has not been sufficient to eliminate completely individ-
ual variation. However, the percentages are fairly constant,
and it is not probable that the addition of data from more
pupils would vary the order of difficulty or seriously change
the weighting of the exercises.

Very poor A M Very good

Fig. 1

In weighting the exercises we have assumed that the
distribution of pupils according to the ability in question will
result in the normal frequency-curve as shown in Fig. 1. The
direction from left to right on CD will be considered positive.
The line MN represents the median. If AP is drawn so that
AMNP is one-fourth the entire surface under the curve, then
AM is known as the P. E. (possible error) .

Reference to Table I shows that Exercise 2 is the easiest
of the eleven exercises used. The score for this exercise is 82.
It is, therefore, 82 — 50, or 32 per cent too easy for the median
pupil. Converting this difference into P. E. 2 values, we find
that Exercise 2 falls at — 1.357 P. E.; that is, 1.357 P. E. to the

1 Information concerning the prices of tests and directions for giving and scoring the
tests may be had by addressing J. H. Minnick, College Hall, University of Pennsylvania,

2 Table XIII of Trabue's Completion Test Language Scale.

PUPILS' ABILITY TO DEMONSTRATE THEOREMS 105

left of the median. Since our zero-point must be arbitrary,
we may select it so as to give convenient numbers for values
of the various exercises. Hence we have selected —2.357 P. E.
as our zero-point. This makes Exercise 2 fall at 1 P. E. above
the zero-point.

Table II indicates the method of determining the value of
each exercise. Thus the first line indicates that 42 per cent
of the pupils solved Exercise 1 correctly, that the exercise is 8

TABLE II

Values Assigned to Each Exercise

Exercise

Per Cent
Correct

Difference

between

50 Per Cent

and Score

P. E.

Value

Distance

in P. E.

above

Zero-Point

Value

42.O
82.O

49-3
27.9
67.7
3*3
56.7
l 3-3
39-3
20.0

34-o

+ 8.0

—32.O

+ 0.7
+ 22. 1
—17.7

+ 18.7
-6.7

+ 36.7
+ IO.7
+30.0
+ I6.0

+ O.299

—'•357
+0.026
+0.869
-0.681
+0.723
— 0.250
+ 1.649

+0.403
+ 1.248
+0.612

2.656

1 .OOO

2 383
3.226
I.676
3.080
2.IO7
4.006
2.760
3.605
2.969

27
IO

2

7

24

3 2
17
3i
21

C

6

7

8

40
28

q

IO

36
3°

II

per cent too difficult for the median pupil and is, therefore,
0.299 P- E. above the median or 2.656 P. E. above our arbitrary
zero-point, and that the value assigned to Exercise 1 is 27.
In order to avoid the decimal point in the values assigned,
0.1 P. E. has been taken as the unit. The values are then
obtained by moving the decimal point one place to the right
in the next to the last column of the table.

Fig. 2 is the linear projection of the eleven exercises.
From these exercises it was necessary to select those which
should constitute the scale. When the tests were given, a

io6

THE SCHOOL REVIEW

record was kept of the time required by each pupil to do the
three exercises assigned to him. This time varied from 5 to
30 minutes. The average time spent by a pupil was about 1 8
minutes. It, therefore, seemed that five exercises were sufficient
for the 15 or 20 minutes 1 which the pupils will be given to
work on the final scale. An examination of Fig. 2 shows that

Exerolses-

1 r.E.

1 —

2 ».«.

t

Fig. 2

3 P.E.

19 11 6 4

■"I—
10

4 P.E.

>l

Exercises 2, 5, 3, and 6 are distributed along the scale at almost
equal intervals. The interval between 6 and 10 is somewhat
smaller, and that between 6 and 8 somewhat larger than the
intervals by which 2, 5, 3, and 6 are separated. Exercise 10
occurs as a theorem in some texts, and the results from the
schools in which the tests were given were clearly affected by
this fact. Hence Exercise 8 was selected as the fifth exercise
of our scale.

TABLE III
Order of Difficulty for the Eight Largest Schools

Schools

Order

1

2

3

4

5

A

2
2
2
2

2

5

2

2

5
3
5

5
5
2

3

5

3
5
3
6

3
3
5
3

6
6
6

3
6
6
6
6

8

B

8

C

8

D

8

E

8

F

8

G

8

H

8

The selection of these five exercises is also justified by the
fact that two fall above, two below, and one almost at the

■This time is to be determined experimentally during the year 1919-20.

PUPILS' ABILITY TO DEMONSTRATE THEOREMS 107

median. Also, if we examine Table I, we find that these
exercises have been given to enough pupils to give almost
constant results. The order of difficulty of these five exercises
for students completing the first two books of geometry and
for pupils completing all of plane geometry has been carefully
examined and has been found to be identical in the two cases.
Table III shows the order of difficulty for the eight largest
schools. Thus for School A the array of exercises from the
easiest to the most difficult is 2, 5, 3, 6, 8. Four of the eight
schools have the same order of difficulty. In each of the other
schools there is one and only one reversal of the order. In no
case is an exercise removed more than one place from the order
given by all of the schools taken together.

Exercises 2, 5, 3, 6, 8, will constitute our test and will be
numbered I, II, III, IV, V, respectively. The exercises in full
follow:

Given: The circle whose center is O, arc AB = arc BC, BM is
perpendicular to AO, BN is perpendicular to OC. Make any additional
drawings that are necessary to prove that BM = BN.

io8

THE SCHOOL REVIEW

ll

-B

F

Given: AB is parallel to CD, and lines GE and FE meet in E.
Make any additional drawings that are necessary to prove that
Z2= Zi+ Z3-

HI

Given: Triangle ABC inscribed in a circle whose center is 0, and OD
perpendicular to CB.

Make any additional drawings that are necessary to prove that Zi= Z2.

PUPILS' ABILITY TO DEMONSTRATE THEOREMS

109

Given: The triangle ABC, AC = BC, D is any point on AB, DE is
perpendicular to BC, DG is perpendicular to AC.

Make any additional drawings necessary to prove that DG + DE = AF.

Given: The circle whose center is 0, chord AC = chord BD, AC and BD
intersect in P, and the line OP is drawn.

Make any additional drawings that are necessary to prove that Z 1 = /.1.

```